• Nuclear Force• General principles• Examples: quantitative nuclear theory;

predictive capability • Realities of high-

performance computing

Nuclear structure III (theory)Witek Nazarewicz (UTK/ORNL)

National Nuclear Physics Summer School 2014William & Mary, VA

The Force

Nuclear forceA realistic nuclear force force:

schematic view

• Nucleon r.m.s. radius ~0.86 fm• Comparable with interaction range• Half-density overlap at max. attarction• VNN not fundamental (more like inter-

molecular van der Waals interaction)• Since nucleons are composite objects,

three-and higher-body forces are expected.

Reid93 is fromV.G.J.Stoks et al., PRC49, 2950 (1994).

AV16 is fromR.B.Wiringa et al., PRC51, 38 (1995).

Repusive core

attraction

There are infinitely many equivalent nuclear potentials!

QCD!

quark-gluon structuresoverlap heavy mesons

OPEP

Nucleon-Nucleon interaction (qualitative analysis)

Bogner, Kuo, Schwenk, Phys. Rep. 386, 1 (2003)

nucleon-nucleon interactions

N3LO: Entem et al., PRC68, 041001 (2003)Epelbaum, Meissner, et al.

Vlow-k unifies NN interactions at low energy

Effective-field theory potentials Renormalization group (RG) evolved

nuclear potentials

The computational cost of nuclear 3-body forces can be greatly reduced by decoupling low-energy parts from high-energy parts, which can then be discarded.

Three-body forces between protons and neutrons are analogous to tidal forces: the gravitational force on the Earth is not just the sum of Earth-Moon and Earth-Sun forces (if one employs point masses for Earth, Moon, Sun)

Recently the first consistent Similarity Renormalization Group softening of three-body forces was achieved, with rapid convergence in helium. With this faster convergence, calculations of larger nuclei are possible!

three-nucleon interactions

The challenge and the prospect: NN force

Beane et al. PRL 97, 012001 (2006) and Phys. Rev. C 88, 024003 (2013)

Ishii et al. PRL 99, 022001 (2007)

Optimizing the nuclear forceinput matters: garbage in, garbage out

A. Ekström et al., Phys. Rev. Lett. 110, 192502 (2013)

• The derivative-free minimizer POUNDERS was used to systematically optimize NNLO chiral potentials

• The optimization of the new interaction NNLOopt yields a χ2/datum ≈ 1 for laboratory NN scattering energies below 125 MeV. The new interaction yields very good agreement with binding energies and radii for A=3,4 nuclei and oxygen isotopes

• Ongoing: Optimization of NN + 3NF

http://science.energy.gov/np/highlights/2014/np-2014-05-e/

• Used a coarse-grained representation of the short-distance interactions with 30 parameters

• The optimization of a chiral interaction in NNLO yields a χ2/datum ≈ 1 for a mutually consistent set of 6713 NN scattering data

• Covariance matrix yields correlation between LECCs and predictions with error bars.

Navarro Perez, Amaro, Arriola, Phys. Rev. C 89, 024004 (2014) and arXiv:1406.0625

Deuteron, Light Nuclei

Binding energy 2.225 MeV

Spin, parity 1+

Isospin 0

Magnetic moment m=0.857 mN

Electric quadrupole moment

Q=0.282 e fm2

produced by tensor force!

Deuteron

short-range

three-body

Nucleon-Nucleon InteractionNN, NNN, NNNN,…, forces

GFMC calculations tell us that:

A=2: many years ago…

3H: 1984 (1% accuracy)• Faddeev• Schroedinger

3He: 1987

4He: 1987

5He: 1994 (n-a resonance)

A=6,7,..12: 1995-2014

Few-nucleon systems(theoretical struggle)

• A first rate theory predicts• A second rate theory forbids• A third rate theory explains after the

factsAlexander I. Kitaigorodskii

Happy the man who has been able to discern the cause of things

Virgil, Georgica

TheoriesModels

Weinberg’s Laws of Progress in Theoretical PhysicsFrom: “Asymptotic Realms of Physics” (ed. by Guth, Huang, Jaffe, MIT Press, 1983)

First Law: “The conservation of Information” (You will get nowhere by churning equations)

Second Law: “Do not trust arguments based on the lowest order of perturbation theory”

Third Law: “You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you’ll be sorry!”

Nuclear structureNuclear reactionsNew standard model

Hot and dense quark-gluon matter

Hadron structure

Applications of nuclear science

Hadron-Nuclear interface

Res

olut

ion

Effe

ctiv

e F

ield

The

ory

DFT

collectivemodels

CI

ab initio

LQCD

quarkmodels

scaleseparation

To explain, predict, use…

How are nuclei made?Origin of elements, isotopes

Modeling the Atomic Nucleus

Theoretical bag of tricks…

The Nuclear Many-Body Problem

coupled integro-differential equations in 3A dimensions

dimension of the problem

How to explain the nuclear landscape from the bottom up? Theory roadmap

11Li100Sn240Pu 298U

Theory of nuclei is demanding

InputForces, operators

• rooted in QCD• insights from EFT• many-body interactions• in-medium renormalization• microscopic functionals• low-energy coupling constants

optimized to data• crucial insights from exotic

nuclei

Many-bodydynamics

• many-body techniqueso direct ab initio schemes o symmetry breaking and

restoration• high-performance computing• interdisciplinary connections

Open channels

• nuclear structure impacted by couplings to reaction and decay channels

• clustering, alpha decay, and fission still remain major challenges for theory

• unified picture of structure and reactions

Illustrative physics examples

Ab initio theory for light nuclei and nuclear matter

Ab initio: QMC, NCSM, CCM,…(nuclei, neutron droplets, nuclear matter)

Quantum Monte Carlo (GFMC) 12C

No-Core Shell Model 14F, 14C

Coupled-Cluster Techniques 17F, 56Ni, 61Ca

Input: • Excellent forces based on the

phase shift analysis and few-body data

• EFT based nonlocal chiral NN and NNN potentials

• SRG-softened potentials based on bare NN+NNN interactions

NN+NNNinteractions

Renormalization

Ab initio input

Many bodymethod

Observables

• Direct comparison with experiment

• Pseudo-data to inform theory

Green’s Function Monte Carlo(imaginary-time method)

Trial wave function

Other methods:• Faddeev-Yakubovsky method• Hyperspherical harmonics method• Coupled-cluster expansion method, exp(S)• Cluster approaches (resonating group method)• No-core shell model• Lattice methods

GFMC: S. Pieper, ANL

1-2% calculations of A = 6 – 12 nuclear energies are possibleexcited states with the same quantum numbers computed

Ab initio: Examples

12C: ground state and Hoyle statestate-of-the-art computing

Epelbaum et al., Phys. Rev. Lett. 109, 252501 (2012). Lattice EFTLahde et al., Phys. Lett. B 732, 110 (2014).

Pieper et al., QMC

Wiringa et al. Phys. Rev. C 89, 024305 (2014); A. Lovato et al., Phys. Rev. Lett. 112, 182502 (2014) The ADLB (Asynchronous Dynamic Load-

Balancing) version of GFMC was used to make calculations of 12C with a complete Hamiltonian (two- and three-nucleon potential AV18+IL7) on 32,000 processors of the Argonne BGP. The computed binding energy is 93.5(6) MeV compared to the experimental value of 92.16 MeV and the point rms radius is 2.35 fm vs 2.33 from experiment.

-91.7(2)

The frontier: neutron-rich calcium isotopesprobing nuclear forces and shell structure in a neutron-rich medium

ISOLTRAP@CERNWienholtz et al, Nature (2013)TITAN@TRIUMF

Gallant et al, PRL 109, 032506 (2012)

CC theoryHagen et al., PRL109, 032502 (2012)

52Ca mass

54Ca mass

54Ca 2+

54Ca: 20 protons, 34 neutrons

RIBF@RIKENSteppenbeck et alNature (2013)

Ab Initio Path to Heavy NucleiBinder et al., arXiv:1312.5685

• The first accurate ab initio cupled cluster calculations for heavy nuclei using SRG-evolved chiral interactions. A number of technical hurdles eliminated

• Many-body calculations up to 132Sn are now possible with controlled uncertainties on the order of 2%

• A first direct validation of chiral Hamiltonians in the regime of heavy nuclei using ab initio techniques.

• Future studies will have to involve consistent chiral Hamiltonians at N3LO considering initial and SRG-induced 4N interactions and provide an exploration of other observables.

Configuration interaction techniques

• light and heavy nuclei• detailed spectroscopy• quantum correlations (lab-system description)

NN+NNNinteractions Renormalization

DiagonalizationTruncation+diagonalization

Monte Carlo

Observables

• Direct comparison with experiment

• Pseudo-data to inform reaction theory and DFT

Matrix elementsfitted to experiment

Input: configuration space + forces Method

Nuclear shell model

Residualinteractioni

One-bodyHamiltonian

• Construct basis states with good (Jz, Tz) or (J,T)• Compute the Hamiltonian matrix• Diagonalize Hamiltonian matrix for lowest eigenstates• Number of states increases dramatically with particle number

• Can we get around this problem? Effective interactions in truncated spaces (P-included, finite; Q-excluded, infinite)

• Residual interaction (G-matrix) depends on the configuration space. Effective charges

• Breaks down around particle drip lines

Microscopic valence-space Shell Model Hamiltonian

G.R. Jansen et al., arXiv:1402.2563

S.K. Bogner et al., arXiv:1402.1407

Coupled Cluster Effective Interaction(valence cluster expansion)

In-medium SRG Effective Interaction

Anomalous Long Lifetime of 14C

Dimension of matrix solved for 8 lowest states ~ 109 Solution took ~ 6 hours on 215,000 cores on Cray XT5 Jaguar at ORNL

Determine the microscopic origin of the suppressed b-decay rate: 3N force

Maris et al., PRL 106, 202502 (2011)

Fusion of Light Nuclei

Ab initio theory reduces uncertainty due to conflicting data The n-3H elastic cross section for 14 MeV neutrons,

important for NIF, was not known precisely enough. Delivered evaluated data with required 5%

uncertainty and successfully compared to measurements using an Inertial Confinement Facility

“First measurements of the differential cross sections for the elastic n-2H and n-3H scattering at 14.1 MeV using an Inertial Confinement Facility”, by J.A. Frenje et al., Phys. Rev. Lett. 107, 122502 (2011)

http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.107.122502

NIF

Computational nuclear physics enables us to reach into regimes where experiments and analytic theory are not possible, such as the cores of fission reactors or hot and dense evolving environments such as those found in inertial confinement fusion environment.

32

NN+NNNinteractions

Density MatrixExpansion

Input

Energy DensityFunctional

Observables

• Direct comparison with experiment

• Pseudo-data for reactions and astrophysics

Density dependentinteractions

Fit-observables• experiment• pseudo data

Optimization

DFT variational principleHF, HFB (self-consistency)

Symmetry breaking

Symmetry restorationMulti-reference DFT (GCM)

Time dependent DFT (TDHFB)

Nuclear Density Functional Theory and Extensions

• two fermi liquids• self-bound• superfluid (ph and pp channels)• self-consistent mean-fields• broken-symmetry generalized product states

Technology to calculate observablesGlobal properties

SpectroscopyDFT Solvers

Functional formFunctional optimization

Estimation of theoretical errors

Mean-Field Theory Density Functional Theory⇒

• mean-field one-body densities⇒

• zero-range local densities⇒

• finite-range gradient terms⇒

• particle-hole and pairing channels

• Has been extremely successful. A broken-symmetry generalized product state does surprisingly good job for nuclei.

Nuclear DFT• two fermi liquids• self-bound• superfluid

Degrees of freedom: nucleonic densities

Mass table

Goriely, Chamel, Pearson: HFB-17Phys. Rev. Lett. 102, 152503 (2009)

dm=0.581 MeV

Cwiok et al., Nature, 433, 705 (2005)

BE differences

Examples: Nuclear Density Functional Theory

Traditional (limited) functionals provide quantitative description

Example: Large Scale Mass Table Calculations

5,000 even-even nuclei, 250,000 HFB runs, 9,060 processors – about 2 CPU hours

Full mass table: 20,000 nuclei, 12M configurations — full JAGUAR

HFB+LN mass table, HFBTHO

Description of observables and model-based extrapolation• Systematic errors (due to incorrect assumptions/poor modeling)• Statistical errors (optimization and numerical errors)

Erler et al., Nature (2012)

How many protons and neutrons can be bound in a nucleus?

Skyrme-DFT: 6,900±500syst

Literature: 5,000-12,000

288~3,000

Erler et al.Nature 486, 509 (2012)

Asymptotic freedom ?

from B. Sherrill

DFTFRIB

current

Quantified Nuclear Landscape

Neutron star crust

Neutron star crust

Astronomical observables

Astronomical observables

Nuclear observables

Nuclear observables

Many-body theory

Many-body theory

Nuclearinteractions

Nuclearinteractions

Nuclear matter equation of stateNuclear matter

equation of state

Microphysics(transport,…)Microphysics(transport,…)

Quest for understanding the neutron-rich matter on Earth and in the Cosmos

RNBfacilities

Gandolfi et al. PRC85, 032801 (2012)

The covariance ellipsoid for the neutron skin Rskin in 208Pb and the radius of a 1.4M⊙ neutron star. The mean values are: R(1.4M⊙ )=12 km and Rskin= 0.17 fm.

J. Erler et al., PRC 87, 044320 (2013)

From nuclei to neutron stars (a multiscale problem)

Major uncertainty: density dependence of the symmetry energy. Depends on T=3/2 three-nucleon forces